From the Wikipedia article Open Primaries in the United States
States with an open presidential primary
Alabama
Arizona (Semi-closed, with primaries open only to unaffiliated or unrepresented voters)
Arkansas
Georgia
Hawaii (Open primary for state, local, and congressional races; caucus system for presidential races.)
Illinois
Massachusetts (All races' primaries open for "unenrolled"/unaffiliated voters only)
Michigan
Minnesota
Mississippi
Missouri
New Hampshire
North Carolina
North Dakota
South Carolina
Tennessee
Texas
Vermont
Virginia
Wisconsin